Lewy's example

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In mathematics, in the field partial differential equations, Lewy's example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions. It removes the hope that the analog of the Cauchy-Kovalevskaya theorem can hold, in the smooth category.

The "example" itself is not explicit, since it employs the Hahn-Banach theorem, but there since have been various explicit examples of the same nature found by Harold Jacobowitz.

The Malgrange-Ehrenpreis theorem states (roughly) that linear partial differential equations with constant coefficients always have at least one solution; Lewy's example shows that this result cannot be extended to linear partial differential equations with polynomial coefficients.

1 The Example
2 Significance for CR manifolds
3 Significance for hypoelliptic PDE
4 References

[edit] The Example

The statement is as follows

On RxC, there exists a smooth complex-valued function F(t,z) such that the differential equation

$\frac{\partial u}{\partial\bar{z}}-iz\frac{\partial u}{\partial t} = F(t,z)$

admits no solution on any open set. (If F is analytic then the Cauchy-Kovalevskaya theorem implies there is a solution.)

Lewy constructs this F using the following result:

On RxC, suppose that u(t,z) is a function satisfying, in a neighborhood of the origin,

$\frac{\partial u}{\partial\bar{z}}-iz\frac{\partial u}{\partial t} = \phi'(t)$

for some C¹ function φ. Then φ must be real-analytic in a (possibly smaller) neighborhood of the origin.

This may be construed as a non-existence theorem by taking φ to be merely a smooth function. Lewy's example takes this latter equation and in a sense translates its non-solvability to every point of RxC. The method of proof uses a Baire category argument, so in a certain precise sense almost all equations of this form are unsolvable.

Mizohata (1962) later found that the even simpler equation

$\frac{\partial u}{\partial x}+ix\frac{\partial u}{\partial y} = F(x,y)$

depending on 2 real variables x and y sometimes has no solutions. This is almost the simplest possible partial differential operator with non-constant coefficients.

[edit] Significance for CR manifolds

A CR manifold comes equipped with a chain complex of differential operators, formally similar to the Dolbeault complex on a complex manifold, called the $\bar{\partial}_b$ -complex. The Dolbeault complex admits a version of the Poincaré lemma. In the language of sheaves, this asserts that the Dolbeault complex is exact. The Lewy example, however, shows that the $\bar{\partial}_b$ -complex is almost never exact.